# Is it known what the necessary and sufficient conditions are for the existence of a “3+1 split” (by means of a foliation) of a (Lorentzian) manifold?

When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to have a way of splitting our spacetime $M$ into a part that we call space and one we call time. This is necessary, for example, in order to write the two-form $F$ that represents the electromagnetic field as $F = E \wedge dt + B$. I know that this is possible if we assume $M$ to be compact, but this implies that there are closed time-like geodesics (I have been told I would not know a proof). What are the necessary and sufficient conditions for such a foliation to exist? I would also appreciate it if you could inform me about books/resources that go into these sorts of questions. I have looked into books about GR, but they don't seem to talk about these sorts of things.

• I tried to improve your title, but had to (subjectively) interpret your question in order to nicely formulate it: Please feel free to roll back my edit and replace the title by something better if you want to. – Danu Feb 3 '16 at 11:32
• I think you can use the Frobenius theorem to show foliation of the manifold into spacelike hypersurfaces – Slereah Feb 3 '16 at 12:26
• Related MO.SE posts: mathoverflow.net/q/198552/13917 , mathoverflow.net/q/214683/1391 and links therein. – Qmechanic Feb 3 '16 at 12:41
• @Slereah as far as I know, Frobenius gives equivalent conditions for a foliation to be integrable. It does not tell you much (anything?) about which spaces would admit certain foliations. – Danu Feb 3 '16 at 13:23
• Thanks guys! The articles that can be found in the links suggested by Qmechanic seem very interesting and give quite weak conditions for such a foliation to exist! I will certainly look into them! – Anonymous Feb 3 '16 at 18:14